Generalized numerical ranges, joint positive definiteness and multiple eigenvalues

Poon, Yiu

doi:10.1090/S0002-9939-97-03781-7

Generalized numerical ranges, joint positive definiteness and multiple eigenvalues
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by Yiu Tung Poon

Proc. Amer. Math. Soc. 125 (1997), 1625-1634

DOI: https://doi.org/10.1090/S0002-9939-97-03781-7

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Abstract:

We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy’s result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust’s result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.

References

J. F. Adams, Peter D. Lax, and Ralph S. Phillips, On matrices whose real linear combinations are non-singular, Proc. Amer. Math. Soc. 16 (1965), 318–322. MR 179183, DOI 10.1090/S0002-9939-1965-0179183-6
Yik-hoi Au-yeung, A simple proof of the convexity of the field of values defined by two Hermitian forms, Aequationes Math. 12 (1975), 82–83. MR 366954, DOI 10.1007/BF01834040
Yik Hoi Au-Yeung, Some results on generalized numerical ranges and related topics, Current trends in matrix theory (Auburn, Ala., 1986) North-Holland, New York, 1987, pp. 23–28. MR 898888
Yik Hoi Au-Yeung and Yiu Tung Poon, A remark on the convexity and positive definiteness concerning Hermitian matrices, Southeast Asian Bull. Math. 3 (1979), no. 2, 85–92. MR 564798
Yik Hoi Au-Yeung and Nam-Kiu Tsing, Some theorems on the generalized numerical ranges, Linear and Multilinear Algebra 15 (1984), no. 1, 3–11. MR 731673, DOI 10.1080/03081088408817574
Yik Hoi Au-Yeung and Nam-Kiu Tsing, An extension of the Hausdorff-Toeplitz theorem on the numerical range, Proc. Amer. Math. Soc. 89 (1983), no. 2, 215–218. MR 712625, DOI 10.1090/S0002-9939-1983-0712625-4
C. A. Berger, Normal dilations, Ph. D. Thesis, Cornell Univ., 1963.
F. Bohnenblust, Joint positiveness of matrices, unpublished manuscript.
Louis Brickman, On the field of values of a matrix, Proc. Amer. Math. Soc. 12 (1961), 61–66. MR 122827, DOI 10.1090/S0002-9939-1961-0122827-1
John Doyle, Analysis of feedback systems with structured uncertainties, Proc. IEE-D 129 (1982), no. 6, 242–250. MR 685109, DOI 10.1049/ip-d.1982.0053
Michael K. H. Fan and André L. Tits, $m$-form numerical range and the computation of the structured singular value, IEEE Trans. Automat. Control 33 (1988), no. 3, 284–289. MR 927847, DOI 10.1109/9.405
S. Friedland and R. Loewy, Subspaces of symmetric matrices containing matrices with a multiple first eigenvalue, Pacific J. Math. 62 (1976), no. 2, 389–399. MR 414597, DOI 10.2140/pjm.1976.62.389
P. R. Halmos, Numerical ranges and normal dilations, Acta Sci. Math. (Szeged) 25 (1964), 1–5. MR 171168
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge University Press, London, 1934.
F. Hausdorff, Der Wertvorrat einer Bilinearform, Math Z. 3 (1919), 314–316.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Chi-Kwong Li, Matrices with some extremal properties, Linear Algebra Appl. 101 (1988), 255–267. MR 941306, DOI 10.1016/0024-3795(88)90153-X
Albert W. Marshall and Ingram Olkin, Inequalities: theory of majorization and its applications, Mathematics in Science and Engineering, vol. 143, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 552278
Yiu Tung Poon, Another proof of a result of Westwick, Linear and Multilinear Algebra 9 (1980), no. 1, 35–37. MR 586669, DOI 10.1080/03081088008817347
Yiu Tung Poon, On the convex hull of the multiform numerical range, Linear and Multilinear Algebra 37 (1994), no. 1-3, 221–223. Special Issue: The numerical range and numerical radius. MR 1313770, DOI 10.1080/03081089408818324
Olga Taussky, Positive-definite matrices, Inequalities (Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965) Academic Press, New York, 1967, pp. 309–319. MR 0220754
R. C. Thompson, Singular values, diagonal elements, and convexity, SIAM J. Appl. Math. 32 (1977), no. 1, 39–63. MR 424847, DOI 10.1137/0132003
O. Toeplitz, Das algebraishe Analogon zu einem Satze von Fejér, Math. Z. 2 (1918), 187–197.
R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra 2 (1975), 311–315. MR 374936, DOI 10.1080/03081087508817074